The Numerical Simulation of General Relativistic Shock Waves by a Locally Inertial Godunov Method Featuring Dynamical Time Dilation

TL;DR

This paper presents a novel numerical method for simulating general relativistic shock waves, successfully modeling phenomena like black hole formation and shock reflections in Einstein's equations for a perfect fluid.

Contribution

It introduces a locally inertial Godunov method with dynamical time dilation, enabling the first numerical simulation of fluid dynamical shock waves in general relativity.

Findings

Resolved secondary reflected shock waves in simulations.

Indicated black hole formation from smooth initial conditions.

First numerical simulation of relativistic fluid shock waves.

Abstract

We introduce what we call a locally inertial Godunov method with dynamical time dilation, and use it to simulate a new one parameter family of general relativistic shock wave solutions of the Einstein equations for a perfect fluid. The forward time solutions resolve the secondary reflected wave (an incoming shock wave) in the Smoller-Temple shock wave model for an explosion into a static singular isothermal sphere. The backward time solutions indicate black hole formation from a smooth underlying solution via collapse associated with an incoming rarefaction wave. As far as we know, this is the first numerical simulation of a fluid dynamical shock wave in general relativity.